Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain

Active and inactive information loss

Jeremy RodgersSection 5 of 16Original PDF ↗Publication DOI ↗

Active and inactive information loss

We now sharpen the picture. Corollary 3.2 tells us that fiber variation of $a_Q$ obstructs. But in practice one usually knows something more specific: one knows which source-level structure the readout has forgotten, and one wants to know whether that particular forgotten structure is what the target depends on. This is the distinction between information that is merely lost and information whose loss has consequences.

Definition 4.1 (Source relation and its target factorization).

Let $r\colon S\to V$ be a map, called a source relation, thought of as recording some physically meaningful attribute of the reduced source state that the readout does not itself determine. We say that $Q$ factors through $(p,r)$ if there exists a map $\Phi_Q\colon T\times V\longrightarrow A_Q \qquad\text{such that}\qquad a_Q(s)=\Phi_Q\bigl(p(s),r(s)\bigr)\quad\text{for all }s\in S.\tag{7}$

Equation (7) says that the readout value together with the relation value suffices to determine the correct answer. It is a modeling hypothesis about how much of the source the target actually consults; the interesting content lies in what one may then conclude.

Definition 4.2 (Activity and inactivity on a fiber).

Assume (7), and let $t\in T$ . The relation $r$ is active for $Q$ on the fiber $F_t$ if there exist $s_0,s_1\in F_t$ with $\Phi_Q\bigl(t,r(s_0)\bigr)\neq\Phi_Q\bigl(t,r(s_1)\bigr).\tag{8}$ It is inactive for $Q$ on $F_t$ if no such pair exists, i.e. if the map $\Phi_Q(t,\cdot)$ is constant on the realized set $r(F_t)=\{\, r(s) : s\in F_t \,\}\subseteq V .\tag{9}$ We say $r$ is globally inactive for $Q$ if it is inactive on every fiber.

Two features of Definition 4.2 carry the weight of the section, and both are easy to get wrong.

First, inactivity is a statement about the realized target values, not about $r$ . The relation $r$ may vary wildly across $F_t$ , $r(F_t)$ may be an enormous subset of $V$ , and $r$ may still be inactive for $Q$ on $F_t$ , provided $\Phi_Q(t,\cdot)$ happens to be constant on that subset. The readout has then forgotten a great deal, and none of it matters for $Q$ . This is the normal situation in effective descriptions and is not a pathology.

Second, activity is evaluated on the realized set $r(F_t)$ , not on all of $V$ . That $\Phi_Q(t,\cdot)$ is nonconstant somewhere on $V$ is irrelevant if the source states in $F_t$ never realize the values at which it varies. The obstruction concerns configurations that actually occur in the fiber.

Theorem 4.3 (Active loss obstructs).

Assume (7). If $r$ is active for $Q$ on some fiber $F_{t}$ , then $Q$ is not exactly solvable from the readout $p$ .

Proof. Let $s_0,s_1\in F_{t}$ realize (8). Since $s_0,s_1\in F_{t}$ we have $p(s_0)=t=p(s_1)$ . By (7), $a_Q(s_0)=\Phi_Q\bigl(p(s_0),r(s_0)\bigr)=\Phi_Q\bigl(t,r(s_0)\bigr) \neq \Phi_Q\bigl(t,r(s_1)\bigr)=\Phi_Q\bigl(p(s_1),r(s_1)\bigr)=a_Q(s_1).$ Thus $s_0,s_1$ satisfy the hypothesis (6) of Corollary 3.2, and $Q$ is not exactly solvable from $p$ . Indeed no map $b\colon T\to A_Q$ is correct at both. ◻

Theorem 4.4 (Inactive loss does not obstruct).

Assume (7). If $r$ is globally inactive for $Q$ , that is, if for every $t\in T$ the map $\Phi_Q(t,\cdot)$ is constant on the realized set $r(F_t)$ , then $Q$ is exactly solvable from the readout $p$ . The unique readout-only solution is $\bar a_Q(t)=\Phi_Q\bigl(t,v_t\bigr) \qquad\text{for any } v_t\in r(F_t).\tag{10}$

Proof. We verify condition (2) of Theorem 3.1. Fix $t\in T$ and let $s,s'\in F_t$ . Then $r(s),r(s')\in r(F_t)$ , and by global inactivity $\Phi_Q(t,\cdot)$ is constant on $r(F_t)$ ; hence $\Phi_Q(t,r(s))=\Phi_Q(t,r(s'))$ . Using (7) and $p(s)=p(s')=t$ , $a_Q(s)=\Phi_Q\bigl(t,r(s)\bigr)=\Phi_Q\bigl(t,r(s')\bigr)=a_Q(s').$ So $a_Q$ is constant on $F_t$ , for every $t$ . By Theorem 3.1, $Q$ is exactly solvable and the unique solution satisfies $\bar a_Q(t)=a_Q(s)$ for any $s\in F_t$ ; since $F_t\neq\emptyset$ we may pick such an $s$ and set $v_t:=r(s)\in r(F_t)$ , giving (10). The right-hand side of (10) is independent of the choice of $v_t\in r(F_t)$ , again by constancy of $\Phi_Q(t,\cdot)$ on $r(F_t)$ . ◻

Definition 4.5 (Structural versus consequential loss).

Let $r\colon S\to V$ be a source relation that does not factor through $p$ ; that is, $r$ is not constant on some fiber. We call this structural information loss: the readout genuinely fails to determine $r$ . If in addition $r$ is active for a target $Q$ on some fiber, we call the loss consequential for $Q$ , or target-active.

Theorem 4.3, Theorem 4.4 should be read with their standing hypothesis kept in view. Both presuppose that the nominated variable $r$ mediates the target, in the precise sense of (7): $a_Q(s)=\Phi_Q(p(s),r(s))$ . Relative to such an $r$ , the two theorems say that activity of $r$ is exactly the condition under which the loss of $r$ obstructs $Q$ , active loss obstructs, and globally inactive loss does not. This is a statement about the chosen mediating variable, not about information loss in the abstract, and it is only as sharp as the choice of $r$ .

The fully general criterion, carrying no such hypothesis, is the one already established in Theorem 3.1, Corollary 3.2: $Q$ is obstructed from the readout precisely when $a_Q$ is nonconstant on some fiber of $p$ . Every statement about activity is a refinement of that criterion, obtained by localizing the fiber variation onto a particular mediating variable, and it adds diagnostic content , which forgotten structure is responsible , rather than logical strength.

With that scoping in place, the moral stands. Structural loss without activity is invisible to $Q$ ; it is what makes coarse descriptions viable rather than defective. It is therefore not a theorem of this framework , and we do not assert , that every noninjective readout requires completion, nor that every lost source distinction must be restored. Whether a given loss must be repaired is a question one can only answer after nominating a target, and the answer may legitimately be no. The examples of Section 9, Section 10, Section 11 include both verdicts for the same readout.

Remark 4.6.

It is worth recording the logical relation between activity of $r$ and descent of $r$ . If $r$ itself descends along $p$ (is fiber-constant), then $r(F_t)$ is a singleton for each $t$ and $r$ is automatically inactive for every target that factors through $(p,r)$ ; the loss is not merely inconsequential but nonexistent. The converse fails: $r$ can vary on fibers , structural loss , while being inactive for $Q$ . So activity implies structural loss, but structural loss does not imply activity, and it is precisely this asymmetry that forbids the inference from “noninjective” to “must be completed.”